TIFR-CAM In-house Symposium 2025

About Symposium

We are pleased to announce "TIFR-CAM In-house Symposium 2025". The objective of this event is to bring together all of us and to communicate our research to a broader audience. We hope that it will give us the opportunity to get introduced to the working areas of each other. We are trying to keep it accessible to as many people as possible and hence everyone is encouraged to attend the talks.

Date

• September 18th (Thursday): 2PM - 5:30PM, and

• September 19th (Friday): 9:45AM - 5:30PM.

Venue: LH-111(TIFR CAM, Bangalore)

Organizers

Schedule

Day 1 September 18, 2025

Day 2 September 19, 2025

Abstracts

Non-classical Solutions to Transport Equations

Abstract: We begin with a review of the classical theory of transport equations in the framework of DiPerna and Lions. We then turn to more recent developments concerning non-classical solutions. In particular, we will focus on the construction of such a solution.

The front tracking method for scalar conservation laws

Abstract: Scalar conservation laws ut + f(u)x = 0 generically develop shocks from initial data, showcasing the limits of classical solution theory via the method of characteristics. Since discontinuities are rampant, it is fruitful to understand solutions of ‘Riemann problems’ – Cauchy problems for initial data  that take exactly two values with a single discontinuity. In this talk, I will present Dafermos’s front tracking method: using Riemann problems as building blocks, we construct piecewise constant approximations on polygonal space-time domains, then prove convergence to entropy solutions for BV initial data. This geometric approach makes shock formation and interaction explicit while rigorously controlling the limiting process through a priori bounds on the spatial total variation.

References
1. Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem by Alberto Bressan.
2. Polygonal approximations of solutions of the initial value problem for a conservation law by Constantine Dafermos. 

Branching Brownian Motion: From Galton’s Extinction Puzzle to the Fisher-KPP Equation

Abstract:  What began as Francis Galton’s Victorian curiosity about the extinction of aristocratic family surnames has evolved into one of modern probability theory’s most elegant bridges to mathematical physics. This talk traces the remarkable mathematical journey from the discrete Galton-Watson process through the continuous realm of Branching Brownian Motion, revealing how probabilistic branching and diffusion naturally give rise to the Fisher-KPP reaction-diffusion equation. Finally, we will explore some recent developments including extremal processes, decorated Poisson point processes, and fixed-point characterizations.

Courant’s Nodal Domain Theorem

Abstract: The eigenfunctions of the Laplace operator describe fundamental modes of vibration, from the shape of a drumhead to the quantum states of a particle. A central object of study is the nodal set, the set of points where an eigenfunction vanishes,and its complement; the nodal domains. In 1923, Richard Courant established a foundational result: the n-th eigenfunction of the Dirichlet Laplacian on a bounded domain has at most n nodal domains. This talk will explore Courant’s celebrated nodal domain theorem, illuminating its elegant proof which combines variational principles with the unique continuation property. We will build prerequisites and results leading up to the theorem. Later we’ll discuss applications and further directions.[1] [2]

References
[1] D. Borthwick. Spectral Theory: Basic Concepts and Applications. Graduate Texts in Mathematics. Springer International Publishing, 2020.
[2] M. Levitin, D. Mangoubi, and I. Polterovich. Topics in Spectral Geometry. Graduate Studies in Mathematics. American Mathematical Society, 2023.

Reproducing Kernel Hilbert Spaces

Abstract: Reproducing Kernel Hilbert Spaces (RKHS) form a bridge between functional analysis, probability, and applied mathematics. They enable pointwise evaluation through the reproducing property, while providing elegant tools to analyze function spaces via kernels. This talk will discuss the construction of RKHS from positive definite kernels and some classical examples. We will explore structural properties, operator-theoretic connections, and their applicationin complex analysis.

Structurally damped semilinear evolution equation for positive operators on Hilbert space

Abstract: In this study, we analyze the following semilinear damped evolution equation associated with a self-adjoint, positive operator L on a separable Hilbert space H with discrete spectrum which, reads as follows:

under different damping conditions, including the undamped (θ = 0), effectively damped (0 < 2θ < σ), critically damped (2θ = σ), non-effectively damped (σ < 2θ < 2σ), and viscoelastic type damped (θ = σ). This article is devoted to examining the decay estimates of solutions to the above linear evolution equation subject to initial Cauchy data with Sobolev regularity. Furthermore, we also observe loss of decay with relaxation of regularity in some case. As an application of the decay estimates, we also demonstrate the global existence (in time) of the solution in certain cases, taking into account polynomial-type nonlinearity.

References
1. Aparajita Dasgupta, Lalit Mohan, and Abhilash Tushir, Structurally
damped semilinear evolution equation for positive operators on Hilbert
space. http://arxiv.org/abs/2507.14581

Metropolis-Hastings Algorithm

Abstract: Metropolis-Hastings algorithm is a simple yet powerful method for sampling from complex probability distributions. In this talk, I will introduce it and explain how it works. 

References
1. Calvetti, D. and Somersalo, E. Introduction to Bayesian Scientific Computing Ten Lectures on Subjective Computing, Surveys and Tutorials in the Applied Mathematical Sciences,Springer, 2007.

Conformal Invariance and stability for an Onofri-type Inequality

Abstract: In this talk, we discuss the quantiative stability bound of an inequality proved by Chang-Gui [CG23], which reads as follows

This is an improvement of classical Trudinger-Moser-Onofri inequality, which has its motivation in the second inequality in Szeg ̈o’s limit theorem for Toeplitz determinants on the circle. We will briefly delve into Minkowski Space time R 1,3 equipped with Lorentzian inner product and see how it comes into picture to prove a conformal invariance of the energy functional
associated to the inequality. This will in turn help us classify the extremizer set of the functional in terms of the conformal group of S2
and finally prove the quadratic stability bound in terms of the distance from the extremizer set. This is a joint work [GK25] with with my advisor Dr. Debabrata Karmakar.

References
[CG23] Sun-Yung Alice Chang and Changfeng Gui. A sharp inequality onthe exponentiation of functions on the sphere. Comm. Pure Appl. Math., 76(6):1303–1326, 2023.

[GK25] Monideep Ghosh and Debabrata Karmakar. Quantitative stability for the conformally invariant Chang-Gui inequality on the exponentiation of functions on the sphere 
https://arxiv.org/abs/2508.19930, 2025.

Spectral Theorem For Normal Operators

Abstract: We all are familiar with spectral theorem for normal matrices . In this talk I will introduce the spectral theorem for normal operators on arbitrary Hilbert spaces . Main ingredient for us will be Projection Valued Measures (measures that assign projections (in certain Hilbert spaces) to measurable subsets) , I will introduce that as well . And we will see some nice applications . Basically I will make sense of following bunch of notations

References
1 . A Course in Functional Analysis , John B. Conway .
2 . C* Algebras and Operator theory , Gerard J. Murphy

On fractional p-Laplace semipositone problems

Abstract: In this talk, we will discuss a class of semipositone problems involving the fractional p-Laplace operator. For the classical p-Laplace case, this problem was previously studied by J. Abrantes Santos, C. O. Alves, and E. Massa (J. Math. Anal. Appl.,527(1), 2023), where the authors employed a non-smooth variational technique to establish the positivity of solutions. However, this method is not readily applicable for s ∈ (0, 1) due to regularity constraints on the critical points of the associated non-smooth energy functional. Therefore, in contrast, we adopt a C1 - variational approach in this work. We consider nonlinearities exhibiting subcritical growth as well as Ambrosetti–Rabinowitz type growth. For a certain range of a parameter, we establish the existence of solutions using the mountain pass theorem. Furthermore, we show that these mountain pass solutions remain uniformly bounded in Lebesgue spaces with respect to the parameter. In addition, we construct an explicit positive radial subsolution, which plays a crucial role in proving the positivity of the solutions.

Explicit inversion of spherical Radon transform in odd dimensions with partial radial data

Abstract: In this talk, we derive an explicit inversion formula for the spherical Radon transform in odd dimensions with partial radial data . We prove that the reconstruction of the unknown function can be reduced to solving an ordinary differential equation. We also provide an analytical solution in some special cases. Finally, we present some numerical simulations that illustrate the efficiency of the proposed inversion formula validating our theoretical results. Our work settles a question posed by Boris Rubin in “Inversion formulae for the spherical mean in odd dimensions and the Euler-Poisson-Darboux equation,” Inverse Problems 24 (2008), no. 2, 025021, 10 pp.

References
1. Pradipta Chatterjee, Venkateswaran P. Krishnan, and Abhilash Tushir. Explicit inversion of spherical radon transform in odd dimensions with partial radial data. (to be communicated), 2025.

Fractional weakly-coupled system with critical nonlinearities

Abstract : Variational problems that exhibit lack of compactness have been a central theme in nonlinear analysis for several decades. In this talk, I will focus on a weakly-coupled, nonhomogeneous system of equations involving the fractional Laplacian operator with critical nonlinearity in the sense of Sobolev embedding on the entire Euclidean space. Particular attention will be given to the role of Palais–Smale sequences in such problems, and I will present a global compactness result. Using Palais–Smale decomposition theorem, I will then discuss results on the existence, uniqueness, and multiplicity of positive solutions for these systems, along with their extension to the p-fractional case. This talk will be based on joint works with Mousomi Bhakta, Patrizia Pucci, Olimpio H. Miyagaki, and Nirjan Biswas [1, 2].

References
[1] Bhakta, Mousomi; Chakraborty, Souptik; Miyagaki,Olimpio H.; Pucci, Patrizia, Fractional elliptic systems with critical nonlinearities. Nonlinearity 34 (2021), no. 11, 7540–7573.
[2] Biswas, Nirjan; Chakraborty, Souptik, On p-fractional weakly-coupled system with critical nonlinearities. Discrete and Continuous Dynamical Systems (2025), Doi: 10.3934/dcds.2025138.